In studying for a qualifying exam, I found a problem asking me to prove:
If $f \in L^1(\mathbb{R})$ has $\int_\mathbb{R} f \neq 0$, then there exists $a \in \mathbb{R}$ with $\int_{(-\infty, a]} f = \int_{[a, \infty)} f$
I'm sure the answer probably has something to do with the fact that $f \in L^1(\mathbb{R})$ requires $|f|$ to be "sufficiently small" as it gets far away from $0$, but I'm really stumped on where to begin with either finding the a relative to f, or showing (i.e., by contradiction) that such an a must exist even without "finding" it.